74 2. Elliptic curves

Exercise 2.12.19. Let e1,e2,e3 be three distinct integers. Show that

Δ = (e1 − e2)(e2 − e3)(e1 − e3) is always even.

Exercise 2.12.20. In this exercise we study the structure of the

quotient G/2G, where G is a finite abelian group.

(1) Let p ≥ 2 be a prime and let G =

Z/peZ,

with e ≥ 1. Prove

that G/2G is trivial if and only if p 2.

(2) Prove that, if G =

Z/2eZ

and e ≥ 1, then G/2G

∼

= Z/2Z.

(3) Finally, let G be an arbitrary finite abelian group. We define

G[2∞] to be the 2-primary component of G, i.e.,

G[2∞]

= {g ∈ G :

2n

· g = 0 for some n ≥ 1}.

In other words, G[2∞] is the subgroup of G formed by those

elements of G whose order is a power of 2. Prove that

G[2∞]

∼

=

Z/2e1

Z ⊕

Z/2e2

Z ⊕ · · · ⊕

Z/2er

Z

for some r ≥ 0 and ei ≥ 1 (here r = 0 means G[2∞] is

trivial). Also show that G/2G

∼

=

(Z/2Z)r.

Exercise 2.12.21. Show that the space

C :

2Y 2 − X2 = 34,

Y

2

−

Z2

= 34

does not have any rational solutions with X, Y, Z ∈ Q. (Hint: modify

the system so there are no powers of 2 in any of the denominators,

then work modulo 8.)

Exercise 2.12.22. For the following elliptic curves, use the method

of 2-descent (as in Proposition 2.10.3 and Example 2.10.4) to find the

rank of E/Q and generators of E(Q)/2E(Q). Do not use Sage:

(1) E :

y2

=

x3

− 14931x + 220590.

(2) E :

y2

=

x3

−

x2

− 6x.

(3) E :

y2

=

x3

− 37636x.

(4) E :

y2

=

x3

−

962x2

+ 148417x. (Hint: use Theorem 2.7.4

first to find a bound on the rank.)

Exercise 2.12.23. Find the rank and generators for the rational

points on the elliptic curve

y2

= x(x + 5)(x + 10).